Jumping spiders can jump up to 40 times their body length
Unsurprisingly, jumping spiders do actually have the ability to jump. This ability comes from their back legs; they rely on their segmented legs and hemolymph (or blood, to you and I) flow to do this.
When a jumping spider wants to jump, they do so by suddenly increasing the hemolymph pressure in their two or four back legs by contracting the muscles in the top half of their bodies. This causes their legs to extend, propelling them forward.
Jumping spiders lack the leg muscles that some hopping insects, like grasshoppers, have.